Global distortion, or blurring, of a picture can arise from various factors such as distortion due to out-of-focus optics and linear translation or shaking of the camera during the exposure time.
Blurring of a digital image may be described by means of a convolution:B0(x)=∫dx′F(x′)h(x−x′)  (1)
where B0(x) is the intensity of the pixel at the address x=(x,y) in the distorted picture, x being a two-dimensional vector, F(x) is the intensity of the pixel x in the undistorted image, and h(x) is the so-called point spread function (PSF) that describes the distortion. The function B0(x) is typically obtained as the output from a digital camera. The PSF for an image distorted by out-of-focus optics, for example, can be described in a first approximation by a function h that is constant inside a circle of radius r and h(x)=0 outside the circle.
Rectifying a distorted image involves determining the function F given the functions B0 and h. The convolution (1) can be Fourier transformed to yield{tilde over (B)}0(q)={tilde over (F)}(q)·{tilde over (h)}(q)  (2)where “{tilde over ( )}” represents the Fourier transform. Hence,
                                          F            ~                    ⁡                      (            q            )                          =                                                            B                ~                            0                        ⁡                          (              q              )                                                          h              ~                        ⁡                          (              q              )                                                          (        3        )            In principle, therefore, F(x) may be obtained from the inverse Fourier transform of {tilde over (B)}(q)/{tilde over (h)}(q). In practice, however, this solution is characterized by a very low signal-to-noise ratio (SNR), due to amplification of noise at frequencies q at which {tilde over (h)}(q) is very small.